In this paper, we justify the semiclassical limit of the Gross--Pitaevskii equation with Dirichlet boundary condition on the three-dimensional upper space under the assumption that the leading-order terms to both initial amplitude and initial phase function are sufficiently small in some high enough Sobolev norms. We remark that the main difficulty of the proof lies in the fact that the boundary layer appears in the leading-order terms of the amplitude functions and the gradient of the phase functions to the WKB expansions of the solutions. In particular, we partially solved the open question proposed in [D. Chiron and F. Rousset, Comm. Math. Phys., 288 (2009), pp. 503--546; C. T. Pham, C. Nore, and M. E. Brachet, Phys. D, 210 (2005), pp. 203--226] concerning the semiclassical limit of the Gross--Pitaevskii equation with Dirichlet boundary condition.

Publication:

SIAM Journal on Mathematical Analysis Vol. 54, Iss. 1

Author:

Guilong Gui

School of Mathematics, Northwest University, Xi’an 710069, China.

E-mail: glgui@amss.ac.cn

Ping Zhang

Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.

E-mail: zp@amss.ac.cn